= Solution
{c}
A <QR decomposition> is $A=QR$, with $Q$ orthogonal and $R$ upper triangular. Orthogonality gives $Q^{\mathsf T}Q=I$, and therefore $\|Qx\|^2=x^{\mathsf T}Q^{\mathsf T}Qx=x^{\mathsf T}x=\|x\|^2$, proving invariance of the <Euclidean norm>.
The unshifted <QR algorithm> starts with $A_0=A$. At step $k$, factor $A_k=Q_kR_k$ and set $A_{k+1}=R_kQ_k$. Then
$$
\boxed{A_{k+1}=Q_k^{\mathsf T}A_kQ_k.}
$$
Thus consecutive matrices are related by <orthogonal similarity>, so they have the same <characteristic polynomial> and <eigenvalues>. Induction proves every iterate has the original <eigenvalues>. Symmetry is also preserved. When the iteration converges to diagonal form, its diagonal entries are the <eigenvalues>; practical implementations use shifts and deflation to improve convergence.
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